Riemann Zeta Function at Even Integers/Lemma
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Lemma
Let $x \in \R$ be such that $\size x < 1$.
Then:
- $\ds \pi x \cot {\pi x} = 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}$
where $\zeta$ denotes the Riemann zeta function.
Proof 1
\(\ds \frac {\sin \pi x} {\pi x}\) | \(=\) | \(\ds \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} }\) | Euler Formula for Sine Function | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {\frac {\sin \pi x} {\pi x} }\) | \(=\) | \(\ds \ln \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {\sin {\pi x} }\) | \(=\) | \(\ds \map \ln {\pi x} + \sum_{k \mathop = 1}^\infty \map \ln {1 - \frac {x^2} {k^2} }\) | Laws of Logarithms | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \pi \frac {\cos {\pi x} } {\sin {\pi x} }\) | \(=\) | \(\ds \frac 1 x + \sum_{k \mathop = 1}^\infty \frac 1 {\paren {1 - \frac {x^2} {k^2} } } \paren {-\frac {2 x} {k^2} }\) | differentiating with respect to $x$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \pi x \cot {\pi x}\) | \(=\) | \(\ds 1 + \sum_{k \mathop = 1}^\infty \frac 1 {\paren {1 - \frac {x^2} {k^2} } } \paren {-\frac {2 x^2} {k^2} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 0}^\infty \paren {\frac {x^2} {k^2} }^n \paren {-\frac {2 x^2} {k^2} }\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 2 \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \paren {\frac {x^2} {k^2} }^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 2 \sum_{n \mathop = 1}^\infty \paren {\sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} } } x^{2 n}\) | interchanging order of summation is valid by Tonelli's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}\) |
$\blacksquare$
Proof 2
From Laurent Series Expansion for Cotangent Function:
- $\ds \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$
where:
- $z \in \C$ such that $\cmod z < 1$
- $\zeta$ is the Riemann Zeta function.
Letting $x \in \R$ replace $z$, and multiplying through by $x$:
\(\ds \pi x \cot \pi x\) | \(=\) | \(\ds x \paren {\frac 1 x - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 2 x \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}\) |
$\blacksquare$